High Voltage Circuit Breakers Vibration Signal Analysis ... · and non-parametric model method, power spectrum estimation based on parametric modeling is an important content of modern

130 Journal of Electronic Systems Volume 5 Number 4 December 2015

High Voltage Circuit Breakers Vibration Signal Analysis based on Power SpectrumEstimation

Fu.ChaoHebei Normal University.Shi Jiazhuang, [email protected]

ABSTRACT: Power spectrum estimation method refers to vibration or random signal power spectrum calculation method, Thepower spectrum analysis is an important part of the signal analysis, can be used to detect the periodic component and thepercentage of the signal. And it has important implications for the diagnosis of mechanical failure.

Keywords: Spectrum Estimation, Vibration Signal, AR Model

Received: 25 May 2015, Revised 26 July 2015, Accepted 2 August 2015

© 2015 DLINE. All Rights Reserved

1. Introduction

Generally speaking, power spectrum estimation method can be divided into classical spectral estimation method and modernspectral estimation method. Classical spectral estimation methods are defined by the finite-length sequence to estimate directly,there are two ways to achieve: (1)Estimate the correlation function first, and then get the power spectrum estimation by the Fouriertransform;(2) Link The power spectrum and the square of the signal amplitude-frequency characteristics. No matter the use ofwhich one way, the common problem is the characteristic estimation of variance is not good, and the estimated value of ups anddowns along the frequency axis is very fierce. With the longer of the data, the phenomenon will be more serious.

Classical spectral estimation method can be divided into direct and indirect methods, direct method is to use a fast Fourieralgorithm (FFT) for a finite number of sample data to obtain a power spectrum of the Fourier transform method, also calledperiodogram; The indirect method is to get the autocorrelation function estimates of sample data firstÿthen performs Fouriertransform of the power spectrum. Since direct method to obtain the power spectrum estimation spectral presence some shortcom-ings such as undulating curves, spectral resolution is not high,several improved algorithms proposed, such as Bartlett Law andWelch method. Modern spectral estimation is mainly made for the classic low resolution spectral estimation variance and poorperformance and other issues, from the way the modern spectral estimation method can be divided into parametric model methodand non-parametric model method, power spectrum estimation based on parametric modeling is an important content of modernpower spectrum estimation. The main parameters of the model method has AR model method, MA and ARMA model methodmodel law.

2. Classic Spectrum Estimation

2.1 Periodogram Method

Journal of Electronic Systems Volume 5 Number 4 December 2015 131

Classical periodogram method (Periodgram) also called direct method, first calculate the Fourier transform of the randomsignal’s N observations x (n), then take the square of the amplitude,and dividing by Sequence of samples’ number N as thesequence x (n) estimates of the true power spectrum, namely:

(1)

When the sample sequence number N tends to infinity, the expected value of the direct method is equal to the power spectrum’s

real value. The spectral estimated resolution is proportional to , and the data truncation and windowing, making that whenthe spectral curve larger the more dramatic ups and downs.

2.2 Direct Method’s Improved AlgorithmsFor the direct method to obtain the power spectrum estimation, when the data length is too large, the spectrum curves undulating;if the data length is too small, the spectral resolution is not high, So people also proposed several improved algorithms, such asBartlett method and Welch method.

2.2.1 Bartlett MethodBartlett average periodogram method is getting N points finite-length sequence x(n) segments seeking re average. Assumingthat x (n) into L segments, and M samples per segment, Thus N = LM, paragraph i sample sequence can be written as:

(3)

(4)

2) Welch Method

Welch method of Bartlett law was amended in two respects: First, select the appropriate window function ω (n), and directlyadding in front of the cycle diagram calculation, so that the resulting cycle Pictured each segment:

(5)

in the formula,

(6)

Is normalization factor. Windowed advantage is that no matter what kind of window functions can make non-negative spectrumestimation; and, when segmented, can overlap between segments, this would reduce the variance.

Corresponding to the sum, and then averaged to obtain an average Is the power spectrum:

(7)

2.3 The indirect methodThe indirect method estimates the correlation function by the sequence x(n) first, Then do Fourier transform of the autocorrelationfunction evaluation, we get the the power spectrum estimation of sequence, x(n) namely:

(8)

2.4 Simulation AnalysisTake a random sequence of noisy is a random white noise, Several methods wereused at the power spectrum estimation, as well as to facilitate comparison, the power spectral density do normalization , asshown in Figure 1.


(a) Time Domain

(b) FFT Spectrum Analysis

(c) Direct method

(d) Indirect method

(e) Bartlett method


(f) Welch method

Figure 1. Comparative sequence x(n) of different power spectrum estimation

Although the Bartlett method does not specify window function, but for finite-length sequence, the equivalent rectangularwindows; Meanwhile to facilitate comparison, Welch method also takes a rectangular window.

From the spectral estimation effect, the resolution of the indirect method is significantly better than the direct method; Comparisonof Figure 1 (e), (f),we can see that Welch method’s spectral estimation’s errer is small than method’s. This is because althoughhave adopted the rectangular window, But in the algorithm, when segments the same sequence, Welch method have overlap

between the paragraphs, but Bartlett method didn’t, Experiments show that Overlapping data length is appropriate, therefore,

the estimated variance characteristics, the former than the latter; If the window function better replaced Hamming window,Blackman window, using the Welch method was relatively smoother curve, as shown in Figure 1.

(a) Hamming window

(b) Gabe Blackman window

Figure 2. Welch power spectrum estimation method

Take a group of scene acquisiting breaker vibration signal, using the indirect method, Welch method to estimate the powerspectrum distribution respectively.


Despite the length of the vibration signal reaches N = 1500, can be counted as long data samples, Spectrum estimation resultsare still not satisfactory. Analysis found that although Welch method does a considerable improvement, but still essentiallyunknown outside the work area data assume that the data is zero, equivalent data windowing, resulting in reduced resolutionand spectral estimation instability; The indirect method is the autocorrelation function of the sequence directly as Fouriertransform, so the impact will be better than the Welch method.

(a)The original signa

(b) FFT spectrum analysis

(c) The indirect method

(d) Welch Act (Gabe Blackman window)

Figure 3. Two power breaker vibration signal spectrum estimation

Modern spectral estimation methods are no longer simply the unknown data is assumed to be observed outside the region andto zero, but first observation data signals to estimate the model parameters, seeking the estimated signal power spectrum inaccordance with the output power of a model, avoided the data hypothetical question of data observation outside the area,


spectrum estimation performance is superior to the classical spectrum estimation. Therefore, the following attempts to adoptmodern spectral estimation methods.

3. AR model of the Power Spectrum Estimation

The power spectrum estimation based on the parametric modeling is an important part of modern power spectrum estimation, Itspurpose is to improve the frequency resolution of the power spectrum estimation. Modern spectral estimation is parametricmodel spectrum estimation method based on random process, It can also be called a parametric model approach or simply modelapproach. Spectrum estimation method based on the general parameters of the model according to the following three steps:

(1) To be determined or estimated random process to select a reasonable model;(2) Model parameter estimation based on known observation data;(3) Model parameter estimation was used to calculate the power spectrum.

Modern spectral estimation model is divided into AR model, MA model and the ARMA model. model, also called moving averagemodel, is the all-zero model; model is also known as auto-regressive moving average model, the “pole-zero” models. or courseof any process can be represented by an infinite order, Even if you choose an appropriate model spectrum estimation procedure,you can always possible to select a sufficiently high order of approximation to the stochastic process modeled. Since accurateestimation of model parameters can be obtained by solving a set of linear equations, and for and model of the power spectrumestimation, accurate estimation of its parameters need to solve a set of higher-order nonlinear equations. Practice shows, ARmodel has a power spectrum for short data analysis, calculation is simple, get a smooth spectral resolution advantages, thereforemodel has been widely used.

3.1 AR Model of the Power Spectrum Estimation FundamentalsA number of stationary random processes can be generated by using white noise excitation linear time-invariant systems. Figure 4, x(n) is composed of an input sequence u(n) excitation output of a causal linear shift invariant discrete-time system H(n).

Figure 4. Parameters of the system model

If the discrete stationary random sequence x (n) can be linear differential equation (8) to describe:

(8)

Where u(n) is a discrete white noise, called {x(n)} is ARMA process.

Formula (8) on both sides were taken Z transform, and assuming b0 = 1, can be obtained by formula (9), that is

(9)

of formula (9)

(10)


(11)

Thereby obtaining the output sequence is a power spectrum

(12)

If b1, b2, ..., bq are all zero, the formula (8) becomes:

(13)

This model is called autoregressive model, referred to as AR, model. Where u(n) is a white noise with variance σ2 sequence, pis the order of the AR model, ak the AR model parameters, k = 1, 2, ..., p. For the AR model, signal x(n) from the previous valueitself several excitation current value and generating a linear combination, and the accurate estimation of AR model parameterscan be obtained by solving a set of linear equations.

Also, the formula (12) becomes:

(14)

That random signal x(n) of the AR model of the power spectrum. It can be seen, to power spectrum estimation, we must obtainthe AR model order p, model parameters a1, a2, ..., ap and σ2.

3.2 Choose the AR Model OrderSelect the appropriate AR model order AR spectral estimation is a key issue, Order p AR model is generally not known inadvance, you need a slightly larger pre-selected value, determined in the recursive process. When using the Levinson recursivealgorithm (Levinson-Durbin), can be given by the low-level to high-end of each set of parameters, and the minimum predictionerror model of the power Pmin (equivalent to white noise sequence variance σ2 is decreasing. Intuitively, p when the predictionerror reaches a specified desired value, or no longer change, then the correct order of the order that is to be selected.

Because prediction error power p is a monotonically decreasing, This value is reduced to just the right number is often hard tochoices, Order selection too low will have a greater bias, and may cause peaks can’t tell; Select the parameters to be estimatedtoo high is because too much performance degradation caused by spectral estimation variance. To this end, there are severaldifferent criteria have been proposed, the following are a few of the more common criteria.

1) Several Common Criteria to Determine the AR Model Ordera) Final Prediction Error Criterion (FPE)

Make FPE(r) for the application of the minimum value of the order of r

(15)

Where r is the order of the prediction error Pr power, N is the number of samples, according to the following formula to calculatePr.

(16)


aj as a model parameter, R(j) of the signal x(n) of the autocorrelation function.

b) Information Criterion (AIC)Make AIC (r) for the application of the minimum value of the order of r

(17)

Wherein the parameters N, Pr same as defined above. When the order r of a start to increase, FPE(r) and the AIC(r) are obtainedat a certain minimum value of the r, r at this time is the most appropriate as the order p.

When the data is shorter, FPE, AIC given the low number of bands, and the results are given in the two is basically the same.When the sample size N tends to infinity, the amount of information FPE(r) and AIC(r) equivalent, when N → ∞, criteria forselecting the correct order of the probability of error is not zero.

c) The Minimum Description Length (MDL)

Rissanen proposed an alternative information criterion -minimum description length (MDL), criterion is based on the process tomake the model corresponding probability distribution probability approaching the maximum possible distribution of the actualprocess of thinking. It is defined as the amount of information.

(18)

MDL statistical information criterion is consistent, experimental results show that the short data, AR order should be selected in

the range will have good results.

Clayton Ulrych and proved that for a short segment, FPE, AIC work was not good.3

d) The Empirical Formula

Engineering frequently using the following formula:

(19)

Where N is the length of the data signal, k is the wavelet packet decomposition level.

According to the formula shows that order of a), b), c) three criteria estimates on the prediction error power Pr, by the formula(16), rP consists x(n) of a sequence of the autocorrelation function r (j) , AR model parameters two parts; According to thedefinition of both, when the sequence x(n) changes in the autocorrelation function r (j) , AR model parameters a (j) values alsochanged; The empirical formula is defined by commonly known, order p by the sequence length N, the number of wavelet packetk decisions and sequence values N, the noise itself is irrelevant; so the four methods, in addition to commonly used empiricalformula, the order p under the other three criteria were randomly change (see Table 1), Wherein i is a random number ofoperations, i = 4.

It should be noted that the above is only a few criteria for order selection provides a basis, for the study to a specific signal, andultimately choose their AR model order, you need to practice to compare the results of multiple.

3.3 Simulation AnalysisTake a sequence of noisy is a random Gaussian white noise, sampling rate

, take points, four methods are solved on AR model parameters p, the results in Table 1.


As can be seen by the sequence x(n) of the expression, because e(n) is a random Gaussian white noise, random valuesgenerated by each operation, so that the sequence x(n) values have changed.

Table 1. Random distribution of each criterion with random Gaussian white noise signal number order

Take another noisy sequence is a noisy square wave, is not randomlydistributed, In the same bandwidth (assuming here are [0,1]) of the premise, change the sample rate fs (equivalent to changingthe sequence length N), the results in Table 2.

Table 2. The following four kinds of sequences of different lengths in order to determine

Found from Table 2, the sequence x(n) by a longer data (N = 2048) becomes short data (N = 32) of the process, the method ofdetermining the order of four substantially decreased. Detailed analysis of the following variation of the order under AIC, FPE,MDL criterion p’s:

(1) AIC criteria and guidelines for the FPE, regardless of size N samples, the results given both are the same, when the numberof samples N is shorter, the order declined rapidly, but compared with the other two methods, the value is relatively high;

(2) For MDL criterion when short data is much smaller than the estimated value given by range, the power spectrumestimation done, the frequency resolution is very poor, or even not affect the resolution of frequency components; For short

data, AR order should be selected in the range will have good results. I try several randomly noisy sequence

analysis results of many experiments show that when changes in the order r range, the result is the estimation ofeach [N/3].

Take the following sequence of noisy , is a Gaussianwhite noise, four methods (AIC, FPE, MDL, commonly used empirical formula) results are given as follows: P1= P2 = 6, P3 = 5, P4= 3; Here Conclusion Based on the above (2) said, adding a case under that criteria when short data, order P3 take [N/3]; Indetermining the order of the AR model r premise Burg method using the estimated power spectrum, spectral estimation resultsshown in Figure 5.


Figure 5. Power spectral analysis method under different criteria Burg


When the data sequence is short, that is N = 32, the order of AIC, FPE standards and guidelines for obtaining similar MDL, whenthe order of P1, P2 , P3 were taken Figure 5 (b), (c) the power spectrum of the frequency curve can be clearly distinguished f1=8Hz, f2 = 12Hz two peaks, this is consistent with the results of (a) the fast Fourier algorithm (FFT) spectrum analysis; the resultsof empirical formula used P4 = 3, the spectral curve is too smooth, the spectral resolution is poor, unable to distinguis f 1= 8Hz,f2 = 12Hz two peaks (Figure 5 (e)), this is because we can see by the empirical formula, the results of the order P4 only by thesignal length N, wavelet packet decomposition level k to decide, regardless of the frequency components of the signal; Figure

5 (d) MDL criterion is also used, the order taking , the effect of the power spectrum estimate (b), (c), as can clearly

distinguish the two frequency peaks; Note that, in order to continue to increase during , the spectrum smoothingsection undulating curve intensified, new peaks appeared, the spectrum changes shown in Figure 6.

When the order r to 13, starting at spectrum spurious peaks (see Figure 6 (c)); continue to increase r, the spectrum hasbeen found in more false peaks, figure 6 (d), (e), (f) shown below, but the pseudo-peak energy relative to the two main peaksappear (f = 8Hz, f = 12Hz Department) is very weak, from the curve was still able to clearly distinguish the two peaks.

Therefore, when a shorter data using MDL criterion, the order can be directly taken [ ], also available at ( , within the

range of ) values, spectral estimation results are basically unaffected.


Figure 6. Different orders of power spectrum estimation

4. Conclusion

Based on the above analysis, although some guidelines have been proposed, but no single criterion can be well suited to avariety of situations, several selection criteria described in the order of only provide a basis for determining the order of the finalsignal is often affected by the specific nature of the study, the analysis of the results of the actual need for several experiments.

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Fu. Chao received the B.S degree in school of Electrical Engineering of Tianjin Hebei University of Technology, China, in 2006and achieved the M.S. degree in 2010. Currently doing a doctorate in Electrical Engineering. He is currently Lecturer in HebeiNormal University, Shi Jiazhuang, China. His research interests are mainly in Signal Processing and Electrical Technology.

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High Voltage Circuit Breakers Vibration Signal Analysis ... · and non-parametric model method, power spectrum estimation based on parametric modeling is an important content of modern